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  4. If the three functions f(x), g(x) and h(x) are such that h(x) = f(x), g(x) and f' (x) ⋅ g' (x) = c, where c is a constant, then (f''(x)/f(x))+ (g''(x)/g(x))+ (2c/f(x).g(x)) is equal to

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Q. If the three functions $f(x), g(x) $ and $ h(x)$ are such that $h(x) = f(x), g(x)$ and $f' (x) \cdot g' (x)$ = c, where $c$ is a constant, then $\frac {f''(x)}{f(x)}+\frac {g''(x)}{g(x)}+\frac {2c}{f(x).g(x)}$ is equal to ____

KCETKCET 2010Continuity and Differentiability

Solution:

Given, $h(x)=f(x) \cdot g(x)$ and $f'(x) \cdot g'(x)=c$
Now, $h'(x)=f'(x) \cdot g(x)+f(x) \cdot g'(x)$
$h''(x)=f''(x) \cdot g(x)+f'(x) \cdot g'(x)$
$+f'(x) \cdot g'(x)+f(x) \cdot g''(x)$
$h''(x)=f''(x) \cdot g(x)+f(x) \cdot g''(x)$
$+2 f'(x) \cdot g'(x)$
$h''(x)=f''(x) \cdot g(x)+f(x) \cdot g''(x)+2 c$...(i)
Now, we find
$\frac{f''(x)}{f(x)}+\frac{g''(x)}{g(x)}+\frac{2 c}{f(x) \cdot g(x)}$
$=\frac{f''(x) \cdot g(x)+g''(x) \cdot f(x)+2 c}{f(x) \cdot g(x)}$
$=\frac{h''(x)}{h(x)}$ [from Eq. (i)]